Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 242

The constrained method achieves the minimum cost function precisely when
the prescribed observational errors are equal to the background error. When the
observational errors are greater than the background error, the background cost
function is overestimated. The observational cost function shows that the overfit
method continues to reduce the observational cost far below the minimum level,
which is a clear indication of strong overfitting. After the second outer-loop, the
constrained method no longer reduces the cost-function as it has reached the
minimum that balances the two gradients.

Cardinalietal. ( 2004 ) showed that J o

is a measure of the degrees of

freedom in the system and that J b

min

is a measure of the degrees of freedom
in the observations. The overfit method reduces

min

J b to zero, thus placing all
weight onto the observations; hence, the degrees of freedom to the observations
is zero. Only the constrained method maintains the proper relationship between the
constraints of the system.

10.5 Discussion

Unconstrained outer-loops in data-space and sequential 3D-Var degrades the solu-
tion and overfits the data. With the proper formulation, weakly nonlinear problems
can be solved with longer time-windows to improve both the number of observa-
tional constraints and length of the trajectory. For problems that are purely linear
(such as when employing the “Representer” method of Bennett ( 2002 )), there is no
need for additional outer-loops because the problem is solved when the inner-loops
converge. For many geophysical applications that are weakly nonlinear, multiple
outer-loops are advantageous. It is important to note that in methods that give greater
weight to the observations (3D-Var, multi-variate optimal interpolation, etc.), careful
consideration must be paid to prevent increments from adding unrealistic structure
to the model in order to fit the observations because it was shown that this structure
leads to severely handicapped predictive skill.
Using a number of quantifiable measures of the assimilation framework to com-
pare the posterior errors of each method, the constrained method preserves the prior
error and does not underestimate the true error. The analysis error of the constrained
method assimilation was consistent with the true error. Although the overfit method
tended to worsen the background initial conditions, it still underestimated the true
analysis error by an average of 7.5 %. Not only does the overfit method provide
an improper minimization, the posterior analysis statistics are invalid. In addition,
the overfit method further underestimated the error in the observations, which is
expected as it gave higher consideration to the noisy observations. The constrained
method was consistent with the true posterior statistics, and as shown by the cost
metrics, provided significant estimate to the degrees of freedom in the system.
When assimilating data in geophysical models, a long time window constrained
by the model dynamics with as many available observations provides a quantifiably